Embedded newform 432.2.bg.a.13.14

• Label: 432.2.bg.a.13.14
• Level: $432$
• Weight: $2$
• Character: 432.13
• Analytic conductor: $3.450$
• Analytic rank: $0$
• Dimension: $840$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.bg (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.44953736732\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(70\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			13.14
Character	\(\chi\)	\(=\)	432.13
Dual form			432.2.bg.a.133.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15556 + 0.815280i) q^{2} +(-1.32179 - 1.11931i) q^{3} +(0.670638 - 1.88421i) q^{4} +(0.762654 - 0.0667236i) q^{5} +(2.43997 + 0.215802i) q^{6} +(1.03298 - 2.83809i) q^{7} +(0.761196 + 2.72407i) q^{8} +(0.494277 + 2.95900i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{4}{9}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15556	+	0.815280i	−0.817104	+	0.576490i
							\(3\)	−1.32179	−	1.11931i	−0.763138	−	0.646236i
\(5\)	0.762654	−	0.0667236i	0.341069	−	0.0298397i								0.0846655	−	0.996409i	\(-0.473018\pi\)
							0.256404	+	0.966570i	\(0.417462\pi\)
\(7\)	1.03298	−	2.83809i	0.390430	−	1.07270i	−0.576376	−	0.817185i	\(-0.695534\pi\)
							0.966806	−	0.255513i	\(-0.0822442\pi\)
\(11\)	0.0316416	−	0.361665i	0.00954029	−	0.109046i	−0.989931	−	0.141548i	\(-0.954792\pi\)
							0.999472	+	0.0325021i	\(0.0103476\pi\)
\(13\)	1.90335	−	1.33274i	0.527893	−	0.369635i	−0.279038	−	0.960280i	\(-0.590016\pi\)
							0.806931	+	0.590645i	\(0.201127\pi\)
\(17\)	−1.11729	+	1.93520i	−0.270982	+	0.469355i	−0.969114	−	0.246614i	\(-0.920682\pi\)
							0.698131	+	0.715970i	\(0.254015\pi\)
\(19\)	0.460428	−	1.71834i	0.105629	−	0.394215i	−0.892786	−	0.450480i	\(-0.851253\pi\)
							0.998416	+	0.0562657i	\(0.0179194\pi\)
\(23\)	−1.54505	−	4.24499i	−0.322165	−	0.885141i	−0.990030	−	0.140858i	\(-0.955014\pi\)
							0.667865	−	0.744283i	\(-0.267208\pi\)
\(29\)	2.87029	−	4.09919i	0.532999	−	0.761201i	−0.458865	−	0.888506i	\(-0.651744\pi\)
							0.991864	+	0.127305i	\(0.0406327\pi\)
\(31\)	−1.19604	+	0.435323i	−0.214815	+	0.0781862i	−0.447186	−	0.894441i	\(-0.647574\pi\)
							0.232371	+	0.972627i	\(0.425352\pi\)
\(37\)	−2.22989	−	8.32205i	−0.366591	−	1.36814i	−0.865251	−	0.501338i	\(-0.832841\pi\)
							0.498660	−	0.866797i	\(-0.333825\pi\)
\(41\)	1.79555	+	0.316604i	0.280417	+	0.0494452i	0.312088	−	0.950053i	\(-0.398972\pi\)
							−0.0316705	+	0.999498i	\(0.510083\pi\)
\(43\)	1.04194	−	11.9094i	0.158894	−	1.81616i	−0.330177	−	0.943919i	\(-0.607108\pi\)
							0.489070	−	0.872244i	\(-0.337336\pi\)
\(47\)	3.65041	+	1.32864i	0.532467	+	0.193802i	0.594239	−	0.804288i	\(-0.297453\pi\)
							−0.0617728	+	0.998090i	\(0.519675\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.13.14	✓	840
16.5	even	4	inner	432.2.bg.a.229.65	yes	840
27.25	even	9	inner	432.2.bg.a.349.65	yes	840
432.133	even	36	inner	432.2.bg.a.133.14	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.13.14	✓	840	1.1	even	1	trivial
432.2.bg.a.133.14	yes	840	432.133	even	36	inner
432.2.bg.a.229.65	yes	840	16.5	even	4	inner
432.2.bg.a.349.65	yes	840	27.25	even	9	inner